President Donald Trump is known to support the bill, which now awaits his signature.

The Senate passed the bill on a 92-5 vote, thereby also avoiding a shutdown of the VA's Choice private-sector program, the Associated Press said. That program was expected to run out of funding by next week, which would have necessitated cutbacks in care to veterans.

The new bill would open up access to private doctors for veterans who don't feel they are getting the care they expect within the VA -- with the approval of a VA health care provider. This would include situations where roadblocks to care exist, such as lengthy wait times or VA medical centers that lack services required by a particular patient.

Some Democrats voiced concerns that the new measure would prove too costly, or that patients might go elsewhere for care that is already provided by the agency.

Sen. Bernie Sanders, I-Vt., was former chairman of the Senate Veterans Affairs Committee. He told the AP that the bill moves the VA "too far in the direction of privatization," adding that it doesn't provide funds to fill more than 30,000 positions at the VA that the Trump administration has left vacant.

"My fear is that this bill will open the door to the draining, year after year, of much needed resources from the VA," he said.

But the White House supported the move, saying the bill would transform the VA "into a high-performing and integrated health care system for the 21st century and provide veterans with more choice in their health care options, whether from VA doctors or from the community."

Trump has also tapped acting VA secretary Robert Wilkie to permanently lead the VA. According to the AP, Democrats say they plan to grill Wilkie on whether he has plans to further privatize the agency.